The 2005 Highlander radio is not integrated as far as I know - it doesn't do anything else, so no controls will be lost. I know what you mean by integrated, when the stereo also has on-screen touch controls for climate control, navigation, etc. This one does not have that.
Budget? Maybe $150 for the stereo (Pioneer sounds good, something with bluetooth to integrate the cell phone), maybe another $100 for the amp, then the sub.
My experience building subs is limited to 1 that I build for home-audio. It's bigger than a mini-fridge and uses a 15" Dayton HF. It's a beast!
Thanks for the comments. I'll check out that other forum.